Decision theory

# Decision theory

Overview
Decision theory in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

is concerned with identifying the value
Value
Value or values may refer to:Concepts of worth:* Value theory – overview of approaches in various disciplines* Value ** Value * Value ** Theory of value ** Value investing...

s, uncertainties
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...

and other issues relevant in a given decision
Decision making
Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...

, its rationality, and the resulting optimal decision
Optimal decision
An optimal decision is a decision such that no other available decision options will lead to a better outcome. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a relative utility to each of them...

. It is closely related to the field of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

as to interactions of agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

with at least partially conflicting interests whose decisions affect each other.
Discussion

Encyclopedia
Decision theory in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

is concerned with identifying the value
Value
Value or values may refer to:Concepts of worth:* Value theory – overview of approaches in various disciplines* Value ** Value * Value ** Theory of value ** Value investing...

s, uncertainties
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...

and other issues relevant in a given decision
Decision making
Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...

, its rationality, and the resulting optimal decision
Optimal decision
An optimal decision is a decision such that no other available decision options will lead to a better outcome. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a relative utility to each of them...

. It is closely related to the field of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

as to interactions of agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....

with at least partially conflicting interests whose decisions affect each other.

## Normative and descriptive decision theory

Most of decision theory is normative
Norm (philosophy)
Norms are concepts of practical import, oriented to effecting an action, rather than conceptual abstractions that describe, explain, and express. Normative sentences imply “ought-to” types of statements and assertions, in distinction to sentences that provide “is” types of statements and assertions...

or prescriptive, i.e., it is concerned with identifying the best decision to take (in practice, there are situations in which "best" is not necessarily the maximal (optimum may also include values in addition to maximum), but within a specific or approximative range), assuming an ideal decision maker who is fully informed, able to compute with perfect accuracy, and fully rational
Rationality
In philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...

. The practical application of this prescriptive approach (how people ought to make decisions) is called decision analysis
Decision analysis
Decision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...

, and aimed at finding tools, methodologies and software to help people make better decisions. The most systematic and comprehensive software tools developed in this way are called decision support system
Decision support system
A decision support system is a computer-based information system that supports business or organizational decision-making activities. DSSs serve the management, operations, and planning levels of an organization and help to make decisions, which may be rapidly changing and not easily specified in...

s.

Since people usually do not behave in ways consistent with axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atic rules, often their own, leading to violations of optimality, there is a related area of study, called a positive
Positive statement
In economics and philosophy, a positive statement concerns what "is", "was", or "will be", and contains no indication of approval or disapproval...

or descriptive discipline, attempting to describe what people will actually do. Since the normative, optimal decision often creates hypotheses for testing against actual behaviour, the two fields are closely linked. Furthermore it is possible to relax the assumptions of perfect information, rationality and so forth in various ways, and produce a series of different prescriptions or predictions about behaviour, allowing for further tests of the kind of decision-making that occurs in practice.

In recent decades, there has been increasing interest in what is sometimes called 'behavioral decision theory' and this has contributed to a re-evaluation of what rational decision-making requires (see for instance Anand, 1993).

### Choice under uncertainty

This area represents the heart of decision theory. The procedure now referred to as expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

was known from the 17th century. Blaise Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

invoked it in his famous wager (see below), which is contained in his Pensées
Pensées
The Pensées represented a defense of the Christian religion by Blaise Pascal, the renowned 17th century philosopher and mathematician. Pascal's religious conversion led him into a life of asceticism, and the Pensées was in many ways his life's work. "Pascal's Wager" is found here...

, published in 1670. The idea of expected value is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an expected value. The action to be chosen should be the one that gives rise to the highest total expected value. In 1738, Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

published an influential paper entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...

to show that expected value theory must be normatively
Norm (philosophy)
Norms are concepts of practical import, oriented to effecting an action, rather than conceptual abstractions that describe, explain, and express. Normative sentences imply “ought-to” types of statements and assertions, in distinction to sentences that provide “is” types of statements and assertions...

wrong. He also gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter, when it is known that there is a 5% chance that the ship and cargo will be lost. In his solution, he defines a utility function and computes expected utility rather than expected financial value (see for a review).

In the 20th century, interest was reignited by Abraham Wald's
Abraham Wald
- See also :* Sequential probability ratio test * Wald distribution* Wald–Wolfowitz runs test...

1939 paper pointing out that the two central procedures of sampling–distribution based statistical-theory, namely hypothesis testing
Statistical hypothesis testing
A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study . In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold...

and parameter estimation
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

, are special cases of the general decision problem. Wald's paper renewed and synthesized many concepts of statistical theory, including loss function
Loss function
In statistics and decision theory a loss function is a function that maps an event onto a real number intuitively representing some "cost" associated with the event. Typically it is used for parameter estimation, and the event in question is some function of the difference between estimated and...

s, risk function
Risk function
In decision theory and estimation theory, the risk function R of a decision rule, δ, is the expected value of a loss function L:...

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....

s, antecedent distributions
Prior probability
In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...

, Bayesian procedures, and minimax
Minimax
Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...

procedures. The phrase "decision theory" itself was used in 1950 by E. L. Lehmann.

The revival of subjective probability theory, from the work of Frank Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...

, Bruno de Finetti
Bruno de Finetti
Bruno de Finetti was an Italian probabilist, statistician and actuary, noted for the "operational subjective" conception of probability...

, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At this time, von Neumann's theory of expected utility proved that expected utility maximization followed from basic postulates about rational behavior.

The work of Maurice Allais
Maurice Allais
Maurice Félix Charles Allais was a French economist, and was the 1988 winner of the Nobel Memorial Prize in Economics "for his pioneering contributions to the theory of markets and efficient utilization of resources."...

and Daniel Ellsberg
Daniel Ellsberg
Daniel Ellsberg, PhD, is a former United States military analyst who, while employed by the RAND Corporation, precipitated a national political controversy in 1971 when he released the Pentagon Papers, a top-secret Pentagon study of U.S. government decision-making in relation to the Vietnam War,...

showed that human behavior has systematic and sometimes important departures from expected-utility maximization. The prospect theory
Prospect theory
Prospect theory is a theory that describes decisions between alternatives that involve risk i.e. where the probabilities of outcomes are known. The model is descriptive: it tries to model real-life choices, rather than optimal decisions.-Model:...

of Daniel Kahneman
Daniel Kahneman
Daniel Kahneman is an Israeli-American psychologist and Nobel laureate. He is notable for his work on the psychology of judgment and decision-making, behavioral economics and hedonic psychology....

and Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...

renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. Kahneman and Tversky found three regularities — in actual human decision-making, "losses loom larger than gains"; persons focus more on changes in their utility–states than they focus on absolute utilities; and the estimation of subjective probabilities is severely biased by anchoring
Anchoring
Anchoring or focalism is a cognitive bias that describes the common human tendency to rely too heavily, or "anchor," on one trait or piece of information when making decisions.-Background:...

.

Castagnoli and LiCalzi (1996), Bordley and LiCalzi (2000) recently showed that maximizing expected utility is mathematically equivalent to maximizing the probability that the uncertain consequences of a decision are preferable to an uncertain benchmark (e.g., the probability that a mutual fund strategy outperforms the S&P 500 or that a firm outperforms the uncertain future performance of a major competitor.). This reinterpretation relates to psychological work suggesting that individuals have fuzzy aspiration levels (Lopes & Oden), which may vary from choice context to choice context. Hence it shifts the focus from utility to the individual's uncertain reference point.

Pascal's Wager
Pascal's Wager
Pascal's Wager, also known as Pascal's Gambit, is a suggestion posed by the French philosopher, mathematician, and physicist Blaise Pascal that even if the existence of God could not be determined through reason, a rational person should wager as though God exists, because one living life...

is a classic example of a choice under uncertainty. It is possible that the reward for belief is infinite (i.e. if God exists and is the sort of God worshiped by evangelical Christians). However, it is also possible that the reward for non-belief is infinite (i.e. if a capricious God exists that rewards us for not believing in God). Therefore, either believing in God or not believing in God, when you include these results, lead to infinite rewards and so we have no decision-theoretic reason to prefer one to the other. (There are several criticisms of the argument.)

### Intertemporal choice

This area is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. If someone received a windfall of several thousand dollars, they could spend it on an expensive holiday, giving them immediate pleasure, or they could invest it in a pension scheme, giving them an income at some time in the future. What is the optimal thing to do? The answer depends partly on factors such as the expected rates of interest
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

and inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...

, the person's life expectancy
Life expectancy
Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...

, and their confidence in the pensions industry. However even with all those factors taken into account, human behavior again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates
Hyperbolic discounting
In behavioral economics, hyperbolic discounting is a time-inconsistent model of discounting.Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to discount the value of the later reward, by a factor that increases with the length of the...

.

### Competing decision makers

Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is more often treated under the label of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, rather than decision theory, though it involves the same mathematical methods. From the standpoint of game theory most of the problems treated in decision theory are one-player games (or the one player is viewed as playing against an impersonal background situation). In the emerging socio-cognitive
Socio-cognitive
Socio-cognitive or sociocognitive describes integrated cognitive and social properties of systems, processes, functions, models, as well as can indicate the branch of science, engineering or technology, such as socio-cognitive research, socio-cognitive interactions.This term is especially used when...

engineering, the research is especially focused on the different types of distributed decision-making in human organizations, in normal and abnormal/emergency/crisis situations.

Signal detection theory is based on decision theory.

### Complex decisions

Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organization that has to make them. In such cases the issue is not the deviation between real and optimal behaviour, but the difficulty of determining the optimal behaviour in the first place. The Club of Rome
Club of Rome
The Club of Rome is a global think tank that deals with a variety of international political issues. Founded in 1968 at Accademia dei Lincei in Rome, Italy, the CoR describes itself as "a group of world citizens, sharing a common concern for the future of humanity." It consists of current and...

, for example, developed a model of economic growth and resource usage that helps politicians make real-life decisions in complex situations.

Observed in many cases is the paradox that more choices may lead to a poorer decision or a failure to make a decision at all. It is sometimes theorized to be caused by analysis paralysis
Analysis paralysis
The term "analysis paralysis" or "paralysis of analysis" refers to over-analyzing a situation, so that a decision or action is never taken, in effect paralyzing the outcome. A decision can be treated as over-complicated, with too many detailed options, so that a choice is never made, rather than...

, real or perceived, or perhaps from rational ignorance
Rational ignorance
Rational ignorance occurs when the cost of educating oneself on an issue exceeds the potential benefit that the knowledge would provide.Ignorance about an issue is said to be "rational" when the cost of educating oneself about the issue sufficiently to make an informed decision can outweigh any...

. A number of researchers including Sheena S. Iyengar
Sheena Iyengar
Sheena Iyengar is the inaugural S.T. Lee Professor of Business in the Management Division at Columbia Business School and the Research Director at the Jerome A. Chazen Institute of International Business. She is known for her research on choice....

and Mark R. Lepper
Mark Lepper
Mark R. Lepper is the Albert Ray Lang Professor of psychology at Stanford University, and a leading theorist in social psychology...

have published studies on this phenomenon. This analysis was popularized by Barry Schwartz in his 2004 book, The Paradox of Choice.

## Alternatives to decision theory

A highly controversial issue is whether one can replace the use of probability in decision theory by other alternatives.

### Probability theory

The Advocates of probability theory point to:
• the work of Richard Threlkeld Cox
Richard Threlkeld Cox
Richard Threlkeld Cox was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations of probability....

for justification of the probability axioms,

• the Dutch book
Dutch book
In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent....

Bruno de Finetti
Bruno de Finetti was an Italian probabilist, statistician and actuary, noted for the "operational subjective" conception of probability...

as illustrative of the theoretical difficulties that can arise from departures from the probability axioms, and

• the complete class theorem
Completeness (statistics)
In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the...

s, which show that all admissible decision rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....

s are equivalent to the Bayesian decision rule for some utility function and some prior distribution (or for the limit of a sequence of prior distributions). Thus, for every decision rule, either the rule may be reformulated as a Bayesian
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

procedure, or there is a (perhaps limiting) Bayesian rule that is sometimes better and never worse.

### Alternatives to probability theory

The proponents of fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...

, possibility theory
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...

, Dempster–Shafer theory and info-gap decision theory
Info-gap decision theory
Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the...

maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success; notably, probabilistic decision theory is sensitive
Sensitivity analysis
Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...

to assumptions about the probabilities of various events, while non-probabilistic rules such as minimax
Minimax
Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...

are robust
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

, in that they do not make such assumptions.

### General criticism

A general criticism of decision theory based on a fixed universe of possibilities is that it considers the "known unknowns", not the "unknown unknowns": it focuses on expected variations, not on unforeseen events, which some argue (as in black swan theory
Black swan theory
The black swan theory or theory of black swan events is a metaphor that encapsulates the concept that The event is a surprise and has a major impact...

) have outsized impact and must be considered – significant events may be "outside model". This line of argument, called the ludic fallacy
Ludic fallacy
The ludic fallacy is a term coined by Nassim Nicholas Taleb in his 2007 book The Black Swan. "Ludic" is from the Latin ludus, meaning "play, game, sport, pastime." It is summarized as "the misuse of games to model real-life situations." Taleb explains the fallacy as "basing studies of chance on the...

, is that there are inevitable imperfections in modeling the real world by particular models, and that unquestioning reliance on models blinds one to their limits.

For instance, a simple model of daily stock market returns may include extreme moves such as Black Monday (1987)
Black Monday (1987)
In finance, Black Monday refers to Monday October 19, 1987, when stock markets around the world crashed, shedding a huge value in a very short time. The crash began in Hong Kong and spread west to Europe, hitting the United States after other markets had already declined by a significant margin...

, but might not model the market breakdowns following the September 11 attacks.